Proceedings - Viện Vật lý

_________________________________________________________________________________Những tiến bộ trong Quang học, Quang tử, Quang phổ và Ứng dụng. 8/2006, Cần Thơ, Việt Nam5. Improvement of the local error methodAnalyzing the procedure of computation of the LEM, we found that it leads to aflexible step-size but that the distribution of the step-sizes along the transmissiondistance is not continuous but step-like. Furthermore, the algorithm leads to thecalculation of many waste steps when somehow the step-size is too large.We propose a modification to this procedure leading to a continuous distribution ofstep-size along the transmission distance. Besides, it avoids the calculation of manywaste steps. This leads to a more efficient and more robust method. The mostimportant point is that, the larger the global error is, the more efficient our method is.The modification focuses on the selection of the step size in Step 5:- If δ > 2ΔG then discard the solution in Step 4 and recalculate with the new step size.- If δ < 2ΔG then take the new step size for the next step.- The new step size is defined by multiplying the present step size by (Δ/ΔG) 1/3This modified local error method (MLEM) ensures that the local error is centeredin ΔG. It prevents from the additional task when the local error increases too far from2ΔG or decreases too far from δG/2. Moreover, it reduces the number of choosingconditions for selecting the step size, leading to a reduction of the computation time.Figure 2 demonstrates the improvement of the performance of the MLEM incomparison with the LEM. These results are obtained by simulating the transmission ofa 10 ps-pulse in a 5 km single-mode fiber. The dispersion of the fiber is 17 ps/km/nm,the nonlinear coefficient is 1.3 W~ 1 km~ 1 and the attenuation is 0.2 dB/km. The timewindow is 320 ps and the number of points 4096. To compare both methods we use theglobal relative error as defined in Ref. [5] :UfUcn(14)UfFigure 2: Comparison of the computational efficiency between LEM and MLEM for a 10ps-pulse duration in a 5 km SMF fiber in three cases : the dispersion effect is dominant(a), the nonlinear effect is dominant;(b) and both effects are comparable (c)where U a is either an analytical solution (when it exists) or a numerical solutionobtained with very small step-sizes, i.e. with very high accuracy. In this paper, U a wasfound by using the S-SSFM with a step-size of 5 cm. U n is the numerical result when258